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   "source": [
    "# softmax回归的简洁实现\n",
    ":label:`sec_softmax_concise`\n",
    "\n",
    "在 :numref:`sec_linear_concise`中，\n",
    "我们发现(**通过深度学习框架的高级API能够使实现**)\n",
    "(~~softmax~~)\n",
    "线性(**回归变得更加容易**)。\n",
    "同样，通过深度学习框架的高级API也能更方便地实现softmax回归模型。\n",
    "本节如在 :numref:`sec_softmax_scratch`中一样，\n",
    "继续使用Fashion-MNIST数据集，并保持批量大小为256。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "7f81001f",
   "metadata": {
    "origin_pos": 2,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "import torch\n",
    "from torch import nn\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "92e395a8",
   "metadata": {
    "origin_pos": 5,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "batch_size = 256\n",
    "train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb2b9199",
   "metadata": {
    "origin_pos": 6
   },
   "source": [
    "## 初始化模型参数\n",
    "\n",
    "如我们在 :numref:`sec_softmax`所述，\n",
    "[**softmax回归的输出层是一个全连接层**]。\n",
    "因此，为了实现我们的模型，\n",
    "我们只需在`Sequential`中添加一个带有10个输出的全连接层。\n",
    "同样，在这里`Sequential`并不是必要的，\n",
    "但它是实现深度模型的基础。\n",
    "我们仍然以均值0和标准差0.01随机初始化权重。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ebf37311",
   "metadata": {
    "origin_pos": 8,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "# PyTorch不会隐式地调整输入的形状。因此，\n",
    "# 我们在线性层前定义了展平层（flatten），来调整网络输入的形状\n",
    "net = nn.Sequential(nn.Flatten(), nn.Linear(784, 10))\n",
    "\n",
    "def init_weights(m):\n",
    "    if type(m) == nn.Linear:\n",
    "        nn.init.normal_(m.weight, std=0.01)\n",
    "\n",
    "net.apply(init_weights);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30a6c3c5",
   "metadata": {
    "origin_pos": 11
   },
   "source": [
    "## 重新审视Softmax的实现\n",
    ":label:`subsec_softmax-implementation-revisited`\n",
    "\n",
    "在前面 :numref:`sec_softmax_scratch`的例子中，\n",
    "我们计算了模型的输出，然后将此输出送入交叉熵损失。\n",
    "从数学上讲，这是一件完全合理的事情。\n",
    "然而，从计算角度来看，指数可能会造成数值稳定性问题。\n",
    "\n",
    "回想一下，softmax函数$\\hat y_j = \\frac{\\exp(o_j)}{\\sum_k \\exp(o_k)}$，\n",
    "其中$\\hat y_j$是预测的概率分布。\n",
    "$o_j$是未规范化的预测$\\mathbf{o}$的第$j$个元素。\n",
    "如果$o_k$中的一些数值非常大，\n",
    "那么$\\exp(o_k)$可能大于数据类型容许的最大数字，即*上溢*（overflow）。\n",
    "这将使分母或分子变为`inf`（无穷大），\n",
    "最后得到的是0、`inf`或`nan`（不是数字）的$\\hat y_j$。\n",
    "在这些情况下，我们无法得到一个明确定义的交叉熵值。\n",
    "\n",
    "解决这个问题的一个技巧是：\n",
    "在继续softmax计算之前，先从所有$o_k$中减去$\\max(o_k)$。\n",
    "这里可以看到每个$o_k$按常数进行的移动不会改变softmax的返回值：\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat y_j & =  \\frac{\\exp(o_j - \\max(o_k))\\exp(\\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))\\exp(\\max(o_k))} \\\\\n",
    "& = \\frac{\\exp(o_j - \\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "\n",
    "在减法和规范化步骤之后，可能有些$o_j - \\max(o_k)$具有较大的负值。\n",
    "由于精度受限，$\\exp(o_j - \\max(o_k))$将有接近零的值，即*下溢*（underflow）。\n",
    "这些值可能会四舍五入为零，使$\\hat y_j$为零，\n",
    "并且使得$\\log(\\hat y_j)$的值为`-inf`。\n",
    "反向传播几步后，我们可能会发现自己面对一屏幕可怕的`nan`结果。\n",
    "\n",
    "尽管我们要计算指数函数，但我们最终在计算交叉熵损失时会取它们的对数。\n",
    "通过将softmax和交叉熵结合在一起，可以避免反向传播过程中可能会困扰我们的数值稳定性问题。\n",
    "如下面的等式所示，我们避免计算$\\exp(o_j - \\max(o_k))$，\n",
    "而可以直接使用$o_j - \\max(o_k)$，因为$\\log(\\exp(\\cdot))$被抵消了。\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\log{(\\hat y_j)} & = \\log\\left( \\frac{\\exp(o_j - \\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))}\\right) \\\\\n",
    "& = \\log{(\\exp(o_j - \\max(o_k)))}-\\log{\\left( \\sum_k \\exp(o_k - \\max(o_k)) \\right)} \\\\\n",
    "& = o_j - \\max(o_k) -\\log{\\left( \\sum_k \\exp(o_k - \\max(o_k)) \\right)}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "我们也希望保留传统的softmax函数，以备我们需要评估通过模型输出的概率。\n",
    "但是，我们没有将softmax概率传递到损失函数中，\n",
    "而是[**在交叉熵损失函数中传递未规范化的预测，并同时计算softmax及其对数**]，\n",
    "这是一种类似[\"LogSumExp技巧\"](https://en.wikipedia.org/wiki/LogSumExp)的聪明方式。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "91c3ac45",
   "metadata": {
    "origin_pos": 13,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "loss = nn.CrossEntropyLoss(reduction='none')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c347cec2",
   "metadata": {
    "origin_pos": 15
   },
   "source": [
    "## 优化算法\n",
    "\n",
    "在这里，我们(**使用学习率为0.1的小批量随机梯度下降作为优化算法**)。\n",
    "这与我们在线性回归例子中的相同，这说明了优化器的普适性。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c4849ef8",
   "metadata": {
    "origin_pos": 17,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "trainer = torch.optim.SGD(net.parameters(), lr=0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2cf8941",
   "metadata": {
    "origin_pos": 20
   },
   "source": [
    "## 训练\n",
    "\n",
    "接下来我们[**调用**] :numref:`sec_softmax_scratch`中(~~之前~~)\n",
    "(**定义的训练函数来训练模型**)。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "5acea90d",
   "metadata": {
    "origin_pos": 21,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
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      "text/plain": [
       "<Figure size 350x250 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "num_epochs = 10\n",
    "d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, trainer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "2f249b8f-52a0-4446-9257-e5a0d22ad65c",
   "metadata": {},
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   ],
   "source": [
    "def predict_ch3(net, test_iter, n=6):  #@save\n",
    "    \"\"\"预测标签（定义见第3章）\"\"\"\n",
    "    for X, y in test_iter:\n",
    "        break\n",
    "    trues = d2l.get_fashion_mnist_labels(y)\n",
    "    preds = d2l.get_fashion_mnist_labels(net(X).argmax(axis=1))\n",
    "    titles = [true +'\\n' + pred for true, pred in zip(trues, preds)]\n",
    "    d2l.show_images(\n",
    "        X[0:n].reshape((n, 28, 28)), 1, n, titles=titles[0:n])\n",
    "\n",
    "predict_ch3(net, test_iter)"
   ]
  },
  {
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   "id": "e09d7d1c",
   "metadata": {
    "origin_pos": 22
   },
   "source": [
    "和以前一样，这个算法使结果收敛到一个相当高的精度，而且这次的代码比之前更精简了。\n",
    "\n",
    "## 小结\n",
    "\n",
    "* 使用深度学习框架的高级API，我们可以更简洁地实现softmax回归。\n",
    "* 从计算的角度来看，实现softmax回归比较复杂。在许多情况下，深度学习框架在这些著名的技巧之外采取了额外的预防措施，来确保数值的稳定性。这使我们避免了在实践中从零开始编写模型时可能遇到的陷阱。\n",
    "\n",
    "## 练习\n",
    "\n",
    "1. 尝试调整超参数，例如批量大小、迭代周期数和学习率，并查看结果。\n",
    "1. 增加迭代周期的数量。为什么测试精度会在一段时间后降低？我们怎么解决这个问题？\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e81d17a2",
   "metadata": {
    "origin_pos": 24,
    "tab": [
     "pytorch"
    ]
   },
   "source": [
    "[Discussions](https://discuss.d2l.ai/t/1793)\n"
   ]
  }
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